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In this work, we study the full set of eigenstates of a $p_x+ip_y$topological superconductor coupled to a particle bath which can be described interms of an integrable Hamiltonian of the Richardson-Gaudin class. The resultsderived in this work also characterise the behaviour of an anisotropic XXZcentral spin model in a external magnetic field since both types of Hamiltonianare know to share the exact same conserved quantities making them formallyequivalent.  We show how by ramping the coupling strength (or equivalently the magneticfield acting in the z-direction on the central spin), each individualeigenstate undergoes a sequence of gain/loss of excitations when crossing thespecific values known as Read-Green points. These features are shown to becompletely predictable, for every one of the $2^N$ eigenstates, using only twointegers obtainable easily from the zero-coupling configuration which definesthe eigenstate in question.  These results provide a complete map of the particle-number sectors(superconductor) or magnetisation sectors (central spin) involved in the largenumber of level-crossings which occur in these systems at the Read-Greenpoints. It further allows us to define quenching protocols which could createstates with remarkably large excitation-number fluctuations.

Read-Green points and level crossings in XXZ central spin models and $p_x+ip_y$ topological superconductors